Three-level Systemsand

Two-photon TransitionsECE 590.01

Quantum Engineering with AtomsGeert Vrijsen

General Rotating Frame

Expectation Values in Rotating Frame

Three-level System in Harmonic Trap(single photon) detuning

Two-photon detuning

Hamiltonians

Atom energy levels

Atom motion in harmonic potential

Atom-light interaction

Interaction Hamiltonian – Schrodinger Picture

Amplitude Polarization

Rotating Frame Hamiltonian

Rotating Frame - Interaction

Relevant Frequency Scales

(Near) resonantly driven:

Raman (Far off resonance):

Rotating Wave Approximation

Motion Dependence – Lamb-Dicke Regime

Lamb-Dicke regime:

For simplicity

Lindblad Dissipation Term – Rotating Frame

Lindblad Dissipation Term – Rotating Frame

Trace-preserving termsPopulation and coherence decay terms

Three-level Hamiltonian Summary

Matrix Form, No Motion (Carrier Transition)

Far Detuned (Raman)

Adiabatic Elimination

Effectively a two-level system!Adiabatically eliminated the excited state

Effective Two-level System

Effective von Neumann equation

Effective Hamiltonian

Effective density matrix

Effective Rabi frequency

Effective detuning

Noteworthy Details

Differential AC Stark shift

Scaling:

Spontaneous emission rate scaling:

For good Raman transitions: • High laser intensity• Large detuning

Differential phase shift